Home MATHEMATICS TOPIC 1: EXPONENTS AND RADICALS ~ MATHEMATICS FORM TWO

TOPIC 1: EXPONENTS AND RADICALS ~ MATHEMATICS FORM TWO

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Exponents
Exponents tell how many times to use a number itself in multiplication. There are different laws that guides in calculations involving exponents. In this chapter we are going to see how these laws are used.
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Indication of power, base and exponent is done as follows:
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Solution:
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To write the expanded form of the following powers:
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Solution
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To write each of the following in power form:
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Soln.
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The Laws of Exponents
List the laws of exponents
First law:Multiplication of positive integral exponent
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Second law: Division of positive integral exponent
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Third law: Zero exponents
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Fourth law: Negative integral exponents
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Verification of the Laws of Exponents
Verify the laws of exponents
First law: Multiplication of positive integral exponent
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Generally,
when we multiply powers having the same base, we add their exponents.
If x is any base and m and n are the exponents, therefore:
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Example 1
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Solution
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If you are to write the expression using the single exponent, for example,(63)4.The expression can be written in expanded form as:
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Generally if a and b are real numbers and n is any integer,
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Example 2
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Example 3
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Example 4
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Generally, (xm)= X(mxn)
Example 5
Rewrite the following expressions under a single exponent for those with identical exponents:
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Second law: Division of positive integral exponent
Example 6
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Example 7
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Therefore,
to divide powers of the same base we subtract their exponents (subtract
the exponent of the divisor from the exponent of the dividend). That
is,
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where x is a real number and x ≠ 0, m and n are integers. m is the exponent of the dividend and n is the exponent of the divisor.
Example 8
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solution
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Third law: Zero exponents
Example 9
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This is the same as:
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If a ≠ 0, then
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Which is the same as:
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Therefore if x is any real number not equal to zero, then X0 = 1,Note that 00is undefined (not defined).
Fourth law: Negative integral exponents
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Also;
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Example 10
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Exercise 1
1. Indicate base and exponent in each of the following expressions:
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2. Write each of the following expressions in expanded form:
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3. Write in power form each of the following numbers by choosing the smallest base:
  1. 169
  2. 81
  3. 10,000
  4. 625
a. 169 b. 81c. 10 000 d. 625
4. Write each of the following expressions using a single exponent:
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5. Simplify the following expressions:
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6. Solve the following equations:
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7. Express 64 as a power with:
  1. Base 4
  2. Base 8
  3. Base 2
Base 4 Base 8 Base 2
8. Simplify the following expressions and give your answers in either zero or negative integral exponents.
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9. Give the product in each of the following:
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10. Write the reciprocal of the following numbers:
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Laws of Exponents in Computations
Apply laws of exponents in computations
Example 11
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Solution
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Re-arranging Letters so that One Letter is the Subject of the Formula
Re-arrange letters so that one letter is the subject of the formula
A formula is a rule which is used to calculate one quantity when other quantities are given. Examples of formulas are:
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Example 14
From the following formulas, make the indicated symbol a subject of the formula:
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Solution
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Transposing a Formulae with Square Roots and Square
Transpose a formulae with square roots and square
Make the indicated symbol a subject of the formula:
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Exercise 3
1. Change the following formulas by making the given letter as the subject of the formula.
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2. Use mathematical tables to find square root of each of the following:
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